Question

question
find the distance between the two points in simplest radical form.
(-8, 3) and (-2, -5)

Explanation:

Step1: Recall the distance formula

The distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \).
Here, \( x_1=-8,y_1 = 3,x_2=-2,y_2=-5 \).

Step2: Substitute the values into the formula

First, calculate \( x_2 - x_1=-2-(-8)=-2 + 8 = 6 \) and \( y_2 - y_1=-5 - 3=-8 \).
Then, substitute these values into the distance formula: \( d=\sqrt{(6)^2+(-8)^2} \).

Step3: Simplify the expression inside the square root

Calculate \( 6^2 = 36 \) and \( (-8)^2=64 \). Then, \( 36 + 64=100 \). So, \( d=\sqrt{100} \)? Wait, no, wait: Wait, \( 6^2=36 \), \( (-8)^2 = 64 \), \( 36+64 = 100 \)? Wait, no, 36+64 is 100? Wait, 36+64=100, but then \( \sqrt{100}=10 \)? Wait, no, wait, let's check again. Wait, \( x_2 - x_1=-2-(-8)=6 \), \( y_2 - y_1=-5 - 3=-8 \). Then \( (x_2 - x_1)^2=6^2 = 36 \), \( (y_2 - y_1)^2=(-8)^2 = 64 \). Then sum is \( 36 + 64=100 \), so square root of 100 is 10. Wait, but that seems too simple. Wait, maybe I made a mistake? Wait, no, let's re - calculate:

Wait, the two points are \((-8,3)\) and \((-2,-5)\). The difference in x - coordinates: \(-2-(-8)=-2 + 8 = 6\). The difference in y - coordinates: \(-5 - 3=-8\). Then, the distance formula is \( \sqrt{(6)^2+(-8)^2}=\sqrt{36 + 64}=\sqrt{100}=10 \). Wait, but 10 is an integer, not a radical. But maybe the problem expects the simplest radical form, and since 100 is a perfect square, the simplest radical form is 10.

Answer:

\( 10 \)